Theorem xorbi12d 1315

Description: Equality property for XOR. (Contributed by Mario Carneiro, 4-Sep-2016.)

Hypotheses

Ref	Expression
xor12d.1	⊢ (φ → (ψ ↔ χ))
xor12d.2	⊢ (φ → (θ ↔ τ))

Assertion

Ref	Expression
xorbi12d	⊢ (φ → ((ψ ⊻ θ) ↔ (χ ⊻ τ)))

Proof of Theorem xorbi12d

Step	Hyp	Ref	Expression
1		xor12d.1	. . . 4 ⊢ (φ → (ψ ↔ χ))
2		xor12d.2	. . . 4 ⊢ (φ → (θ ↔ τ))
3	1, 2	bibi12d 312	. . 3 ⊢ (φ → ((ψ ↔ θ) ↔ (χ ↔ τ)))
4	3	notbid 285	. 2 ⊢ (φ → (¬ (ψ ↔ θ) ↔ ¬ (χ ↔ τ)))
5		df-xor 1305	. 2 ⊢ ((ψ ⊻ θ) ↔ ¬ (ψ ↔ θ))
6		df-xor 1305	. 2 ⊢ ((χ ⊻ τ) ↔ ¬ (χ ↔ τ))
7	4, 5, 6	3bitr4g 279	1 ⊢ (φ → ((ψ ⊻ θ) ↔ (χ ⊻ τ)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ⊻ wxo 1304

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177  df-xor 1305

This theorem is referenced by:  hadbi123d  1382  cadbi123d  1383